Descriptive Statistics Stephen E. Brock, Ph.D., NCSP California - - PDF document

Stephen E. Brock, Ph.D., NCSP EDS 250 Descriptive Statistics 1

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Descriptive Statistics

Stephen E. Brock, Ph.D., NCSP California State University, Sacramento

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Descriptive Statistics

Describes (or summarizes) data. Describes quantitatively (with numbers or graphs) how a particular characteristic is distributed among one or more groups of people. No generalizations beyond the sample represented by the data are made by descriptive statistics.

 However, if your data reflects an entire population, then the

data are considered to be population parameters.

 On the other hand, if your data represents a population

sample, then the data is considered a statistic that describes a

• sample. Inferential statistics are required to determine if the

samples statistics can be generalized back to the population.

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Group Activity: Prepare to teach the following concepts.

What is the “mean” of a data set? What is the “standard deviation” of a data set? What are derived or “standard scores?” What is the “bell shaped curve?”

Stephen E. Brock, Ph.D., NCSP EDS 250 Descriptive Statistics 2

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Preface: Preparing Data for Analysis

Scoring standardized test.

 Follow manual instructions  Have someone else double check 25% of the

protocols

Scoring self-developed measures.

 Establish reliability 5

Preface: Conducting Data for Analysis

Be careful and take your time. Work with a partner who can check your work. Break-up data entry sessions to make sure you avoid the errors caused by fatigue.

 Don’t try to get it all done in one sitting  Double check the work of others (e.g., research

assistants).

 Assuming, accurate data entry, there is no such a

thing as “bad data.”

Develop a coding system and set up a database. The structure of the database will depend upon the type of data being used.

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Preface: Identify Scale of Measurement

Scale Properties E.G.

Nominal (to name) Data represents qualitative or equivalent categories (not numerical). Eye color, Gender, Race or ethnicity (could be a word in database, but…). Mode Ordinal (to order) Numerically ranked, but has no implication about how far apart ranks are. Grades (always a number in the database). Mode, Median Interval (equal) Numerical value indicates rank and meaningfully reflects relative distance between points on a scale Temperature (always a number in the database). Mode, Median, Mean Ratio (equal) Has all the properties of an interval scale, and in addition has a true zero point. Length, Weight (always a number in the database). Mode, Median, Mean

Determines what descriptive statistic will be used

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Preface: Data Entry

Give each participant a subject # Data is generally placed in columns Codes for categorical and nominal data are determined.

 Including group membership (usually

coded as a group number).

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Preface Coding Descriptive Data

Develop a way to code each of the following

• variables. Remember only nominal data can be

coded with words or letters, all other data must be quantified.

 Eye color  Grades  IQ scores  Weight  Gender  Art skill level  Temperature  Length  Ethnicity  Race results

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A Sample Data Base

S# EC Gd IQ Wt Sx Art Eth RR 1 Bn 4 100 97 1 10 1 1 2 Bn 3 105 65 1 3 1 2 3 Bn 4 130 200 2 5 3 3 4 Bl 4 111 99 2 7 4 4 5 H 1 90 43 1 9 2 5 6 Bn 65 55 1 2 5 6 7 G 2 117 67 1 4 6 7 8 H 2 100 87 2 6 7 8 9 Bl 2 89 96 1 8 3 9 10 Bn 4 85 45 2 1 3 10

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Preface: Coding Experimental Data

Develop a way to code each of the following

• variables. Remember only nominal data can

be coded with words or letters, all other data must be quantified.

 Group membership (ADHD Int, v ADHD Hyp v Bipolar Type 1)  Hyperactivity

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A Sample Data Base

S# Group T-Score 1 1 60 2 1 50 3 1 55 4 2 71 5 2 78 6 1 65 7 2 88 8 2 70 9 2 89 10 2 85 S# Group T-Score 11 1 50 12 1 55 13 2 79 14 2 65 15 1 50 16 1 61 17 1 58 18 2 65 19 2 88 20 1 50 S# Group T-Score 21 3 79 22 3 65 23 3 80 24 3 71 25 3 78 26 3 65 27 3 88 28 3 70 29 3 61 30 3 90

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Types of Descriptive Statistics

Univariate (single variable data set summaries)

 Measures of Central Tendency (location)  Measures of Variability (dispersion)  Measures of Shape (symmetry of the normal curve)  Measures of Relative Position (rank, standard score)

Bivariate (two data sets)

 Measures of Relationship

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Measures of Central Tendency

Mode

 Determined by looking at a set of scores and seeing

which occurs most frequently.

 Not typically used, however, it is the only appropriate

statistic for nominal data.

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Measures of Central Tendency

Median

 The point above and below which 50% of the scores

are found.

 Unlike the mode, may not be one of the obtained

results.

 e.g., if there are an even number of scores the median is the point halfway between the two middle scores  i.e., in “13, 25, 27, 45” the median = 26

 When an extreme score is a part of the data set the

median will not be the best estimate of the group’s performance.

 Appropriate for use when the data is ordinal. 15

Group Activity: Prepared to teach the following concepts.

What is the “mean” of a data set?

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Measures of Central Tendency

Mean

 Most frequently used measure of central tendency  The arithmetic average of the scores  Appropriate for use when the data is interval or ratio. 17

Measures of Central Tendency

For example…

 Compute measures of central tendency for the

following data set of math standard scores

 96, 96, 97, 99, 100, 101, 102, 104, 195

 Mode = 96  Median = 100  Mean = 110.6

 What does each measure of central tendency tell you

about the data set?

 Mode = most frequently obtained score  Median = middle point of obtained range of scores  Mean = when a data set includes one or more extreme scores the mean will reflect the average performance of the group as a whole, but not the most typical result.

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Measures of Variability

Range

 The difference between the highest and the lowest

score.

 A quick estimate of variability.

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Measures of Variability

Variance

 The amount of spread among the scores  In a data set (35, 25, 30, 40, 30) with a mean of 32

the variance is obtained by doing the following:

 35-32 = 3  25-32 = -7  30-32 = -2  40-32 = 8  30-32 = -2  Because the sum of these scores is 0, to estimate the variance each number is squared (9+49+4+64+4 = 130)  130/5 (the number of cases) = 26  Mathematically, to say the variance is 26 is not a problem, but do we typically deal with squared units (do we ask a clerk if 100 squared $ is enough)?

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Group Activity: Prepared to teach the following concepts.

What is the “standard deviation” of a data set?

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Measures of Variability

Standard Deviation (SD)

 The square root of the variance returns the

variance to the metric of the obtained score.

 The most practical estimate of variability.  Small SD indicates the scores are close together

(little variability)

 What will this distribution “look” like?

 Large SD indicates the scores are far apart (large

variability)

 What will this distribution “look” like?

 If the distribution is normal, over 99% of the

• btained scores will fall with in + or – 3 standard

deviations from the mean.

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Once upon a time . . .

One sunny Saturday morning, down by the banks, of the Hankie Pankie. A group

• f Woodchucks congregated with the

intent of competing in the international Woodchuck, wood-chucking competition. These are the results of the pounds of wood-chucked in one 24 hour period.

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Compute Mean, Median, Mode, Range, Variance, Standard Deviation

Larry - 95 lbs Charles - 100lbs Vic - 125 lbs Bertha - 85 lbs Bunny - 90 lbs Chauncy - 95 lbs

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Measures of Central Tendency and Variability

Mode: Most frequently occurring Median: Point above/below which 50% of scores occur Mean: The average of the scores Range: The difference between the highest and lowest scores Variance: Amount of spread among the scores. Standard Deviation: Measure of Variability of a distribution of test scores.

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Group Activity: Prepared to teach the following concepts.

What is the “bell shaped curve?”

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Measures of Shape

When population or sample scores on a particular characteristic are graphed, the shape of the “normal curve” resembles a bell. The majority of scores fall in the middle (near the mean), and a few scores fall at the extreme ends of the curve. The height of a “normal curve” will be determined by the variability of the scores.

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Measures of Shape

The Normal (or bell shaped) Curve If a variable is normally distributed it falls in a normal or bell shaped curve. Characteristics

 50% of scores are above/below the

mean

 Mean, median, mode have the same

value (a reason for looking at all three)

 Most scores are near the mean. Fewer

scores are away from the mean.

 The same number of scores are found +

and - a standard deviation from the mean.

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The Bell-Shaped Curve

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Measures of Shape

Skewed Distributions Assumptions that form the basis for parametric inferential statistical analyses require a normal (or near normal) distributions.

 Non-parametric statistics are used when the

distribution is not “normal.”

Skewed distributions are asymmetrical (either positively or negatively). Examination of the mean, median, and mode will tell you if a distribution is skewed or not.

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Symmetrical and Asymmetrical Distributions

Negatively Skewed Distribution Positively Skewed Distribution

A distribution with no skew (e.g. a normal distribution) is symmetrical A negatively skewed distribution has a longer tail to the left A positively skewed distribution has a longer tail to the right

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Activity

Interpret these descriptive statistics Assuming these data are normally distributed, what do you think their respective bell shaped curves might look like?

Group N Mean Hyp. T-score Median T-Score SD Range ADHD Hyp 10 72 70 14.17 69 to 90 ADHD Int 10 55 50 22.79 45 to 61 Bipolar Typ 1 10 77 71 14.98 70 to 99

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Group Activity: Prepared to teach the following concepts.

What are derived or “standard scores?”

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Measures of Relative Position

Percentile Rank

 Percent of scores that fall at or below a given score  Appropriate for ordinal data, typically computed for

interval data.

 Ranks are much closer together at the center of the

distribution.

Standard Scores

 A derived score that reflects how far a score is from a

reference point (typically the mean)

 Z-Scores, # of SDs from the mean.  T Scores, a z score that has been transformed in some

way

 Difference between all scores are equal regardless of

location on distribution.

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Measures of Relative Position

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Measures of Relationship

Correlation

 Determine whether and to what degree a

relationship exists between two or more quantifiable variables

 Degree of relationship is expressed via the

correlation coefficient.

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Measures of Relationship

Scatter Plots

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Activity

Teaching Descriptive Statistics



Mean



Standard Deviation



Standard Score



Bell Shaped Curve

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April 23

Data Analysis: Inferential Statistics



Read Educational Research Chapter 19.

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Portfolio Element #10 Due: Identify resources that will assist you in analyzing data. These resources do not necessarily need to be CSUS resources. Portfolio entries could include student descriptions of the data analysis resources

• identified. Alternatively, any descriptive

handout(s) describing how to locate/use a given resource may be included.